Axioms

19 pages, 2403 KiB

Open AccessReview

Mathematical Modeling of Rogue Waves: A Survey of Recent and Emerging Mathematical Methods and Solutions

by Sergio Manzetti

Axioms 2018, 7(2), 42; https://doi.org/10.3390/axioms7020042 - 20 Jun 2018

Cited by 12 | Viewed by 6300

Anomalous waves and rogue events are closely associated with irregularities and unexpected events occurring at various levels of physics, such as in optics, in oceans and in the atmosphere. Mathematical modeling of rogue waves is a highly active field of research, which has [...] Read more.

Anomalous waves and rogue events are closely associated with irregularities and unexpected events occurring at various levels of physics, such as in optics, in oceans and in the atmosphere. Mathematical modeling of rogue waves is a highly active field of research, which has evolved over the last few decades into a specialized part of mathematical physics. The applications of the mathematical models for rogue events is directly relevant to technology development for the prediction of rogue ocean waves and for signal processing in quantum units. In this survey, a comprehensive perspective of the most recent developments of methods for representing rogue waves is given, along with discussion of the devised forms and solutions. The standard nonlinear Schrödinger equation, the Hirota equation, the MMT equation and other models are discussed and their properties highlighted. This survey shows that the most recent advancement in modeling rogue waves give models that can be used to establish methods for the prediction of rogue waves in open seas, which is important for the safety and activity of marine vessels and installations. The study further puts emphasis on the difference between the methods and how the resulting models form the basis for representing rogue waves in various forms, solitary or with a wave background. This review has also a pedagogic component directed towards students and interested non-experts and forms a complete survey of the most conventional and emerging methods published until recently. Full article

(This article belongs to the Special Issue Applications of Differential Geometry)

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16 pages, 285 KiB

Open AccessArticle

Neutrosophic Quadruple BCK/BCI-Algebras

by Young Bae Jun, Seok-Zun Song, Florentin Smarandache and Hashem Bordbar

Axioms 2018, 7(2), 41; https://doi.org/10.3390/axioms7020041 - 18 Jun 2018

Cited by 29 | Viewed by 3314

Abstract

The notion of a neutrosophic quadruple

B C K / B C I

-number is considered, and a neutrosophic quadruple

B C K / B C I

-algebra, which consists of neutrosophic quadruple

B C K / B C I

-numbers, is constructed. [...] Read more.

The notion of a neutrosophic quadruple

B C K / B C I

-number is considered, and a neutrosophic quadruple

B C K / B C I

-algebra, which consists of neutrosophic quadruple

B C K / B C I

-numbers, is constructed. Several properties are investigated, and a (positive implicative) ideal in a neutrosophic quadruple

B C K

-algebra and a closed ideal in a neutrosophic quadruple

B C I

-algebra are studied. Given subsets A and B of a

B C K / B C I

-algebra, the set

N Q (A, B)

, which consists of neutrosophic quadruple

B C K / B C I

-numbers with a condition, is established. Conditions for the set

N Q (A, B)

to be a (positive implicative) ideal of a neutrosophic quadruple

B C K

-algebra are provided, and conditions for the set

N Q (A, B)

to be a (closed) ideal of a neutrosophic quadruple

B C I

-algebra are given. An example to show that the set

{\tilde{0}}

is not a positive implicative ideal in a neutrosophic quadruple

B C K

-algebra is provided, and conditions for the set

{\tilde{0}}

to be a positive implicative ideal in a neutrosophic quadruple

B C K

-algebra are then discussed. Full article

(This article belongs to the Special Issue Neutrosophic Multi-Criteria Decision Making)

17 pages, 436 KiB

Open AccessArticle

Efficient BEM-Based Algorithm for Pricing Floating Strike Asian Barrier Options (with MATLAB^® Code)

by Alessandra Aimi, Lorenzo Diazzi and Chiara Guardasoni

Axioms 2018, 7(2), 40; https://doi.org/10.3390/axioms7020040 - 15 Jun 2018

Cited by 4 | Viewed by 4027

Abstract

This paper aims to illustrate how SABO (Semi-Analytical method for Barrier Option pricing) is easily applicable for pricing floating strike Asian barrier options with a continuous geometric average. Recently, this method has been applied in the Black–Scholes framework to European vanilla barrier options [...] Read more.

This paper aims to illustrate how SABO (Semi-Analytical method for Barrier Option pricing) is easily applicable for pricing floating strike Asian barrier options with a continuous geometric average. Recently, this method has been applied in the Black–Scholes framework to European vanilla barrier options with constant and time-dependent parameters or barriers and to geometric Asian barrier options with a fixed strike price. The greater efficiency of SABO with respect to classical finite difference methods is clearly evident in numerical simulations. For the first time, a user-friendly MATLAB^® code is made available here. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)

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30 pages, 444 KiB

Open AccessReview

What Do You Mean by “Nonlinear Eigenvalue Problems”?

by Raffaele Chiappinelli

Axioms 2018, 7(2), 39; https://doi.org/10.3390/axioms7020039 - 9 Jun 2018

Cited by 16 | Viewed by 5586

Abstract

A nonlinear eigenvalue problem is generally described by an equation of the form

F (λ, x) = 0

, where

F (λ, 0) = 0

for all

λ

, and contains by definition two unknowns: the eigenvalue [...] Read more.

A nonlinear eigenvalue problem is generally described by an equation of the form

F (λ, x) = 0

, where

F (λ, 0) = 0

for all

λ

, and contains by definition two unknowns: the eigenvalue parameter

λ

and the “nontrivial” vector(s) x corresponding to it. The nonlinear dependence of F can be in either of them (and of course in both), and also the research in this area seems to follow two quite different directions. In this review paper, we try to collect some points of possible common interest for both fields. Full article

20 pages, 253 KiB

Open AccessArticle

Some Summation Theorems for Generalized Hypergeometric Functions

by Mohammad Masjed-Jamei and Wolfram Koepf

Axioms 2018, 7(2), 38; https://doi.org/10.3390/axioms7020038 - 8 Jun 2018

Cited by 12 | Viewed by 4857

Abstract

Essentially, whenever a generalized hypergeometric series can be summed in terms of gamma functions, the result will be important as only a few such summation theorems are available in the literature. In this paper, we apply two identities of generalized hypergeometric series in [...] Read more.

Essentially, whenever a generalized hypergeometric series can be summed in terms of gamma functions, the result will be important as only a few such summation theorems are available in the literature. In this paper, we apply two identities of generalized hypergeometric series in order to extend some classical summation theorems of hypergeometric functions such as Gauss, Kummer, Dixon, Watson, Whipple, Pfaff–Saalschütz and Dougall formulas and also obtain some new summation theorems in the sequel. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications)

2 pages, 154 KiB

Open AccessEditorial

Introduction to Special Issue: New Trends in Fuzzy Set Theory and Related Items

by Humberto Bustince, Javier Fernandez and Esteban Induráin

Axioms 2018, 7(2), 37; https://doi.org/10.3390/axioms7020037 - 5 Jun 2018

Viewed by 3041

Abstract

We focus on the articles recently published in the special issue of Axioms devoted to “New Trends in Fuzzy Set Theory and Related Items”. Full article

(This article belongs to the Special Issue New Trends in Fuzzy Set Theory and Related Items)

28 pages, 407 KiB

Open AccessReview

Line Integral Solution of Differential Problems

by Luigi Brugnano and Felice Iavernaro

Axioms 2018, 7(2), 36; https://doi.org/10.3390/axioms7020036 - 1 Jun 2018

Cited by 90 | Viewed by 5339

Abstract

In recent years, the numerical solution of differential problems, possessing constants of motion, has been attacked by imposing the vanishing of a corresponding line integral. The resulting methods have been, therefore, collectively named (discrete) line integral methods, where it is [...] Read more.

In recent years, the numerical solution of differential problems, possessing constants of motion, has been attacked by imposing the vanishing of a corresponding line integral. The resulting methods have been, therefore, collectively named (discrete) line integral methods, where it is taken into account that a suitable numerical quadrature is used. The methods, at first devised for the numerical solution of Hamiltonian problems, have been later generalized along several directions and, actually, the research is still very active. In this paper we collect the main facts about line integral methods, also sketching various research trends, and provide a comprehensive set of references. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)

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7 pages, 213 KiB

Open AccessArticle

Quantiles in Abstract Convex Structures

by Marta Cardin

Axioms 2018, 7(2), 35; https://doi.org/10.3390/axioms7020035 - 28 May 2018

Viewed by 3294

Abstract

In this short paper, we aim at a qualitative framework for modeling multivariate decision problems where each alternative is characterized by a set of properties. To this extent, we consider convex spaces as underlying universes and make use of lattice operations in convex [...] Read more.

In this short paper, we aim at a qualitative framework for modeling multivariate decision problems where each alternative is characterized by a set of properties. To this extent, we consider convex spaces as underlying universes and make use of lattice operations in convex spaces to formalize the notion of quantiles. We also put in evidence that many important models of decision problems can be viewed as convex spaces-based models. Several properties of aggregation operators are translated into this general setting, and independence and invariance are used to provide axiomatic characterizations of quantiles. Full article

(This article belongs to the Special Issue New Trends in Fuzzy Set Theory and Related Items)

10 pages, 249 KiB

Open AccessArticle

Pre-Metric Spaces Along with Different Types of Triangle Inequalities

by Hsien-Chung Wu

Axioms 2018, 7(2), 34; https://doi.org/10.3390/axioms7020034 - 24 May 2018

Cited by 3 | Viewed by 4285

Abstract

The

T_{1}

-spaces induced by the pre-metric spaces along with many forms of triangle inequalities are investigated in this paper. The limits in pre-metric spaces are also studied to demonstrate the consistency of limit concept in the induced topologies. [...] Read more.

The

T_{1}

-spaces induced by the pre-metric spaces along with many forms of triangle inequalities are investigated in this paper. The limits in pre-metric spaces are also studied to demonstrate the consistency of limit concept in the induced topologies. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications)

17 pages, 407 KiB

Open AccessArticle

Decision-Making with Bipolar Neutrosophic TOPSIS and Bipolar Neutrosophic ELECTRE-I

by Muhammad Akram, Shumaiza and Florentin Smarandache

Axioms 2018, 7(2), 33; https://doi.org/10.3390/axioms7020033 - 15 May 2018

Cited by 61 | Viewed by 5068

Abstract

Technique for the order of preference by similarity to ideal solution (TOPSIS) and elimination and choice translating reality (ELECTRE) are widely used methods to solve multi-criteria decision making problems. In this research article, we present bipolar neutrosophic TOPSIS method and bipolar neutrosophic ELECTRE-I [...] Read more.

Technique for the order of preference by similarity to ideal solution (TOPSIS) and elimination and choice translating reality (ELECTRE) are widely used methods to solve multi-criteria decision making problems. In this research article, we present bipolar neutrosophic TOPSIS method and bipolar neutrosophic ELECTRE-I method to solve such problems. We use the revised closeness degree to rank the alternatives in our bipolar neutrosophic TOPSIS method. We describe bipolar neutrosophic TOPSIS method and bipolar neutrosophic ELECTRE-I method by flow charts. We solve numerical examples by proposed methods. We also give a comparison of these methods. Full article

(This article belongs to the Special Issue Neutrosophic Multi-Criteria Decision Making)

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5 pages, 239 KiB

Open AccessArticle

Varieties of Coarse Spaces

by Igor Protasov

Axioms 2018, 7(2), 32; https://doi.org/10.3390/axioms7020032 - 14 May 2018

Cited by 3 | Viewed by 3389

Abstract

A class

M

of coarse spaces is called a variety if

M

is closed under the formation of subspaces, coarse images, and products. We classify the varieties of coarse spaces and, in particular, show that if a variety

M

contains an unbounded metric [...] Read more.

A class

M

of coarse spaces is called a variety if

M

is closed under the formation of subspaces, coarse images, and products. We classify the varieties of coarse spaces and, in particular, show that if a variety

M

contains an unbounded metric space then

M

is the variety of all coarse spaces. Full article

(This article belongs to the Collection Topological Groups)

36 pages, 552 KiB

Open AccessArticle

Final Value Problems for Parabolic Differential Equations and Their Well-Posedness

by Ann-Eva Christensen and Jon Johnsen

Axioms 2018, 7(2), 31; https://doi.org/10.3390/axioms7020031 - 9 May 2018

Cited by 4 | Viewed by 3370

Abstract

This article concerns the basic understanding of parabolic final value problems, and a large class of such problems is proved to be well posed. The clarification is obtained via explicit Hilbert spaces that characterise the possible data, giving existence, uniqueness and stability of [...] Read more.

This article concerns the basic understanding of parabolic final value problems, and a large class of such problems is proved to be well posed. The clarification is obtained via explicit Hilbert spaces that characterise the possible data, giving existence, uniqueness and stability of the corresponding solutions. The data space is given as the graph normed domain of an unbounded operator occurring naturally in the theory. It induces a new compatibility condition, which relies on the fact, shown here, that analytic semigroups always are invertible in the class of closed operators. The general set-up is evolution equations for Lax–Milgram operators in spaces of vector distributions. As a main example, the final value problem of the heat equation on a smooth open set is treated, and non-zero Dirichlet data are shown to require a non-trivial extension of the compatibility condition by addition of an improper Bochner integral. Full article

18 pages, 8405 KiB

Open AccessArticle

Lyapunov Functions to Caputo Fractional Neural Networks with Time-Varying Delays

by Ravi Agarwal, Snezhana Hristova and Donal O’Regan

Axioms 2018, 7(2), 30; https://doi.org/10.3390/axioms7020030 - 9 May 2018

Cited by 3 | Viewed by 3703

Abstract

One of the main properties of solutions of nonlinear Caputo fractional neural networks is stability and often the direct Lyapunov method is used to study stability properties (usually these Lyapunov functions do not depend on the time variable). In connection with the Lyapunov [...] Read more.

One of the main properties of solutions of nonlinear Caputo fractional neural networks is stability and often the direct Lyapunov method is used to study stability properties (usually these Lyapunov functions do not depend on the time variable). In connection with the Lyapunov fractional method we present a brief overview of the most popular fractional order derivatives of Lyapunov functions among Caputo fractional delay differential equations. These derivatives are applied to various types of neural networks with variable coefficients and time-varying delays. We show that quadratic Lyapunov functions and their Caputo fractional derivatives are not applicable in some cases when one studies stability properties. Some sufficient conditions for stability of equilibrium of nonlinear Caputo fractional neural networks with time dependent transmission delays, time varying self-regulating parameters of all units and time varying functions of the connection between two neurons in the network are obtained. The cases of time varying Lipschitz coefficients as well as nonLipschitz activation functions are studied. We illustrate our theory on particular nonlinear Caputo fractional neural networks. Full article

(This article belongs to the Special Issue Fractional Differential Equations)

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37 pages, 440 KiB

Open AccessArticle

A Survey on the Mathematical Foundations of Axiomatic Entropy: Representability and Orderings

by María-Jesús Campión, Cristina Gómez-Polo, Esteban Induráin and Armajac Raventós-Pujol

Axioms 2018, 7(2), 29; https://doi.org/10.3390/axioms7020029 - 8 May 2018

Cited by 4 | Viewed by 3494

Abstract

Different abstract versions of entropy, encountered in science, are interpreted in the light of numerical representations of several ordered structures, as total-preorders, interval-orders and semiorders. Intransitivities, other aspects of entropy as competitive systems, additivity, etc., are also viewed in terms of representability of [...] Read more.

Different abstract versions of entropy, encountered in science, are interpreted in the light of numerical representations of several ordered structures, as total-preorders, interval-orders and semiorders. Intransitivities, other aspects of entropy as competitive systems, additivity, etc., are also viewed in terms of representability of algebraic structures endowed with some compatible ordering. A particular attention is paid to the problem of the construction of an entropy function or their mathematical equivalents. Multidisciplinary comparisons to other similar frameworks are also discussed, pointing out the mathematical foundations. Full article

13 pages, 281 KiB

Open AccessArticle

Yukawa Potential, Panharmonic Measure and Brownian Motion

by Antti Rasila and Tommi Sottinen

Axioms 2018, 7(2), 28; https://doi.org/10.3390/axioms7020028 - 1 May 2018

Cited by 3 | Viewed by 4625

Abstract

This paper continues our earlier investigation, where a walk-on-spheres (WOS) algorithm for Monte Carlo simulation of the solutions of the Yukawa and the Helmholtz partial differential equations (PDEs) was developed by using the Duffin correspondence. In this paper, we investigate the foundations behind [...] Read more.

This paper continues our earlier investigation, where a walk-on-spheres (WOS) algorithm for Monte Carlo simulation of the solutions of the Yukawa and the Helmholtz partial differential equations (PDEs) was developed by using the Duffin correspondence. In this paper, we investigate the foundations behind the algorithm for the case of the Yukawa PDE. We study the panharmonic measure, which is a generalization of the harmonic measure for the Yukawa PDE. We show that there are natural stochastic definitions for the panharmonic measure in terms of the Brownian motion and that the harmonic and the panharmonic measures are all mutually equivalent. Furthermore, we calculate their Radon–Nikodym derivatives explicitly for some balls, which is a key result behind the WOS algorithm. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications)

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13 pages, 316 KiB

Open AccessArticle

Subordination Properties for Multivalent Functions Associated with a Generalized Fractional Differintegral Operator

by Hanaa M. Zayed, Mohamed Kamal Aouf and Adela O. Mostafa

Axioms 2018, 7(2), 27; https://doi.org/10.3390/axioms7020027 - 24 Apr 2018

Cited by 6 | Viewed by 3884

Abstract

Using of the principle of subordination, we investigate some subordination and convolution properties for classes of multivalent functions under certain assumptions on the parameters involved, which are defined by a generalized fractional differintegral operator under certain assumptions on the parameters involved. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications)

11 pages, 261 KiB

Open AccessArticle

Quotient Structures of BCK/BCI-Algebras Induced by Quasi-Valuation Maps

by Seok-Zun Song, Hashem Bordbar and Young Bae Jun

Axioms 2018, 7(2), 26; https://doi.org/10.3390/axioms7020026 - 23 Apr 2018

Cited by 6 | Viewed by 3370

Abstract

Relations between I-quasi-valuation maps and ideals in

B C K / B C I

-algebras are investigated. Using the notion of an I-quasi-valuation map of a

B C K / B C I

-algebra, the quasi-metric space is induced, and several [...] Read more.

Relations between I-quasi-valuation maps and ideals in

B C K / B C I

-algebras are investigated. Using the notion of an I-quasi-valuation map of a

B C K / B C I

-algebra, the quasi-metric space is induced, and several properties are investigated. Relations between the I-quasi-valuation map and the I-valuation map are considered, and conditions for an I-quasi-valuation map to be an I-valuation map are provided. A congruence relation is introduced by using the I-valuation map, and then the quotient structures are established and related properties are investigated. Isomorphic quotient

B C K / B C I

-algebras are discussed. Full article

23 pages, 953 KiB

Open AccessArticle

On the Analysis of Mixed-Index Time Fractional Differential Equation Systems

by Kevin Burrage, Pamela Burrage, Ian Turner and Fanhai Zeng

Axioms 2018, 7(2), 25; https://doi.org/10.3390/axioms7020025 - 17 Apr 2018

Cited by 4 | Viewed by 4780

Abstract

In this paper, we study the class of mixed-index time fractional differential equations in which different components of the problem have different time fractional derivatives on the left-hand side. We prove a theorem on the solution of the linear system of equations, which [...] Read more.

In this paper, we study the class of mixed-index time fractional differential equations in which different components of the problem have different time fractional derivatives on the left-hand side. We prove a theorem on the solution of the linear system of equations, which collapses to the well-known Mittag–Leffler solution in the case that the indices are the same and also generalises the solution of the so-called linear sequential class of time fractional problems. We also investigate the asymptotic stability properties of this class of problems using Laplace transforms and show how Laplace transforms can be used to write solutions as linear combinations of generalised Mittag–Leffler functions in some cases. Finally, we illustrate our results with some numerical simulations. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)

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12 pages, 302 KiB

Open AccessArticle

New Definitions about A I -Statistical Convergence with Respect to a Sequence of Modulus Functions and Lacunary Sequences

by Ömer Kişi, Hafize Gümüş and Ekrem Savas

Axioms 2018, 7(2), 24; https://doi.org/10.3390/axioms7020024 - 13 Apr 2018

Cited by 2 | Viewed by 4337

Abstract

In this paper, using an infinite matrix of complex numbers, a modulus function and a lacunary sequence, we generalize the concept of

I

-statistical convergence, which is a recently introduced summability method. The names of our new methods are

A^{I}

-lacunary statistical [...] Read more.

In this paper, using an infinite matrix of complex numbers, a modulus function and a lacunary sequence, we generalize the concept of

I

-statistical convergence, which is a recently introduced summability method. The names of our new methods are

A^{I}

-lacunary statistical convergence and strongly

A^{I}

-lacunary convergence with respect to a sequence of modulus functions. These spaces are denoted by

S_{θ}^{A} (I, F)

and

N_{θ}^{A} (I, F),

respectively. We give some inclusion relations between

S^{A} (I, F),

S_{θ}^{A} (I, F)

and

N_{θ}^{A} (I, F)

. We also investigate Cesáro summability for

A^{I}

and we obtain some basic results between

A^{I}

-Cesáro summability, strongly

A^{I}

-Cesáro summability and the spaces mentioned above. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications)

13 pages, 300 KiB

Open AccessArticle

Interval Neutrosophic Sets with Applications in BCK/BCI-Algebra

by Young Bae Jun, Seon Jeong Kim and Florentin Smarandache

Axioms 2018, 7(2), 23; https://doi.org/10.3390/axioms7020023 - 9 Apr 2018

Cited by 40 | Viewed by 3659

Abstract

For

i, j, k, l, m, n \in {1, 2, 3, 4}

, the notion of [...] Read more.

For

i, j, k, l, m, n \in {1, 2, 3, 4}

, the notion of

(T (i, j), I (k, l), F (m, n))

-interval neutrosophic subalgebra in

B C K / B C I

-algebra is introduced, and their properties and relations are investigated. The notion of interval neutrosophic length of an interval neutrosophic set is also introduced, and related properties are investigated. Full article

(This article belongs to the Special Issue Neutrosophic Multi-Criteria Decision Making)

12 pages, 800 KiB

Open AccessArticle

Special Numbers and Polynomials Including Their Generating Functions in Umbral Analysis Methods

by Yilmaz Simsek

Axioms 2018, 7(2), 22; https://doi.org/10.3390/axioms7020022 - 1 Apr 2018

Cited by 14 | Viewed by 4215

Abstract

In this paper, by applying umbral calculus methods to generating functions for the combinatorial numbers and the Apostol type polynomials and numbers of order k, we derive some identities and relations including the combinatorial numbers, the Apostol-Bernoulli polynomials and numbers of order [...] Read more.

In this paper, by applying umbral calculus methods to generating functions for the combinatorial numbers and the Apostol type polynomials and numbers of order k, we derive some identities and relations including the combinatorial numbers, the Apostol-Bernoulli polynomials and numbers of order k and the Apostol-Euler polynomials and numbers of order k. Moreover, by using p-adic integral technique, we also derive some combinatorial sums including the Bernoulli numbers, the Euler numbers, the Apostol-Euler numbers and the numbers

y_{1} (n, k; λ)

. Finally, we make some remarks and observations regarding these identities and relations. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications)

25 pages, 718 KiB

Open AccessArticle

Cross Entropy Measures of Bipolar and Interval Bipolar Neutrosophic Sets and Their Application for Multi-Attribute Decision-Making

by Surapati Pramanik, Partha Pratim Dey, Florentin Smarandache and Jun Ye

Axioms 2018, 7(2), 21; https://doi.org/10.3390/axioms7020021 - 24 Mar 2018

Cited by 16 | Viewed by 4746

Abstract

The bipolar neutrosophic set is an important extension of the bipolar fuzzy set. The bipolar neutrosophic set is a hybridization of the bipolar fuzzy set and neutrosophic set. Every element of a bipolar neutrosophic set consists of three independent positive membership functions and [...] Read more.

The bipolar neutrosophic set is an important extension of the bipolar fuzzy set. The bipolar neutrosophic set is a hybridization of the bipolar fuzzy set and neutrosophic set. Every element of a bipolar neutrosophic set consists of three independent positive membership functions and three independent negative membership functions. In this paper, we develop cross entropy measures of bipolar neutrosophic sets and prove their basic properties. We also define cross entropy measures of interval bipolar neutrosophic sets and prove their basic properties. Thereafter, we develop two novel multi-attribute decision-making strategies based on the proposed cross entropy measures. In the decision-making framework, we calculate the weighted cross entropy measures between each alternative and the ideal alternative to rank the alternatives and choose the best one. We solve two illustrative examples of multi-attribute decision-making problems and compare the obtained result with the results of other existing strategies to show the applicability and effectiveness of the developed strategies. At the end, the main conclusion and future scope of research are summarized. Full article

(This article belongs to the Special Issue Neutrosophic Multi-Criteria Decision Making)

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16 pages, 500 KiB

Open AccessArticle

Graphs in an Intuitionistic Fuzzy Soft Environment

by Sundas Shahzadi and Muhammad Akram

Axioms 2018, 7(2), 20; https://doi.org/10.3390/axioms7020020 - 23 Mar 2018

Cited by 15 | Viewed by 5193

Abstract

In this research article, we present a novel framework for handling intuitionistic fuzzy soft information by combining the theory of intuitionistic fuzzy soft sets with graphs. We introduce the notion of certain types of intuitionistic fuzzy soft graphs including neighbourly edge regular intuitionistic [...] Read more.

In this research article, we present a novel framework for handling intuitionistic fuzzy soft information by combining the theory of intuitionistic fuzzy soft sets with graphs. We introduce the notion of certain types of intuitionistic fuzzy soft graphs including neighbourly edge regular intuitionistic fuzzy soft graphs and strongyl edge irregular intuitionistic fuzzy soft graphs. We illustrate these novel concepts by several examples, and investigate some of their related properties. We present an application of intuitionistic fuzzy soft graph in a decision-making problem and also present our methods as an algorithm that is used in this application. Full article

(This article belongs to the Special Issue New Trends in Fuzzy Set Theory and Related Items)

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Axioms, Volume 7, Issue 2 (June 2018) – 23 articles

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